A-stability of Runge-Kutta methods for systems with additive noise
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Numerical stability of both explicit and implicit Runge-Kutta methods for solving ordinary differential equations with an additive noise term is studied. The concept of numerical stability of deterministic schemes is extended to the stochastic case, and a stochastic analogue of Dahlquist'sA-stability is proposed. It is shown that the discretization of the drift term alone controls theA-stability of the whole scheme. The quantitative effect of implicitness uponA-stability is also investigated, and stability regions are given for a family of implicit Runge-Kutta methods with optimal order of convergence.
1980 AMS Subject Classification65L20 (primary) 60H10,34F05 65L07 93E15
Keywords and phrasesNumerical stability Runge-Kutta methods implicit methods stochastic differential equations stochastic stability
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- 5.J. M. C. Clark, and R. J. Cameron,The maximum rate of convergence of discrete approximations for stochastic differential equations, in B. Grigelionis (ed.),Stochastic differential systems, Lecture Notes in Control and Information Systems, 25, Springer Verlag, Berlin, 1980.Google Scholar
- 8.R. Janssen,Diskretisierung Stochasticher Differentialgleichungen, Preprint Nr. 51, FB Mathematik, Universität Kaiserslautern, 1982.Google Scholar
- 9.P. E. Kloeden and E. Platen,The Numerical Solution of Stochastic Differential Equations, Springer Verlag, Berlin, 1991.Google Scholar
- 13.G. N. Mil'shtein,The Numerical Integration of Stochastic Differential Equations (in Russian), Urals University Press, Sverdlovsk, 1988.Google Scholar
- 15.W. P. Petersen,Stability and accuracy of simulations for stochastic differential equations, IPS Research Report No. 90-02, ETH-Zentrum, Zürich, January 1990.Google Scholar
- 18.J. M. Sancho, M. San Miguel, S. L. Katz and J. D. Gunton,Analytical and numerical studies of computational noise, Phys. Rev. A 26, 1589–1609 (1982).Google Scholar